- #1

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## Homework Statement

Prove that

*lim (1/x) x --> 0*does not exist, i.e., show that

*lim (1/x) x --> 0 = l*is false for every number

*l*.

## Homework Equations

*0 < |x-a| < d*

|f(x) - l| < E

|f(x) - l| < E

## The Attempt at a Solution

The strange thing is, the first time through I got the same solution as Spivak, but looking over it again the logic seems downright wrong. Here's the solution, verbatim:

The functionf(x) = 1/xcannot approach a limit at 0, since it becomes arbitrarily large near 0. In fact, no matter whatd > 0may be, there is somexsatisfying0 < |x| < d,but, namely, any1/x > |l| + Ex < min(d, 1/(|l| + E)). Any suchxdoes not satisfy|(1/x) - l| < E.

Where is he getting the bold portion from?

I write

*|1/x - l| < E*

|1/x| - |l| < |1/x - l| < E

|1/x| - |l| < E

|1/x| < E + |l|

|1/x| - |l| < |1/x - l| < E

|1/x| - |l| < E

|1/x| < E + |l|

How is he getting greater-than?